| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Compare Newton-Raphson with linear interpolation |
| Difficulty | Moderate -0.8 This is a routine Further Maths F1 question testing standard numerical methods procedures. Part (i) requires showing a sign change (straightforward substitution) and applying linear interpolation formula once. Part (ii) involves differentiating a polynomial/rational function and applying Newton-Raphson once. All steps are algorithmic with no problem-solving insight required—easier than average A-level questions. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09d Newton-Raphson method |
\begin{enumerate}
\item (i)
\end{enumerate}
$$f ( x ) = x - 4 - \cos ( 5 \sqrt { x } ) \quad x > 0$$
(a) Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [2.5, 3.5]\\[0pt]
(b) Use linear interpolation once on the interval [2.5, 3.5] to find an approximation to $\alpha$, giving your answer to 2 decimal places.\\
(ii)
$$\operatorname { g } ( x ) = \frac { 1 } { 10 } x ^ { 2 } - \frac { 1 } { 2 x ^ { 2 } } + x - 11 \quad x > 0$$
(a) Determine $\mathrm { g } ^ { \prime } ( x )$.
The equation $\mathrm { g } ( x ) = 0$ has a root $\beta$ in the interval [6,7]\\
(b) Using $x _ { 0 } = 6$ as a first approximation to $\beta$, apply the Newton-Raphson procedure once to $\mathrm { g } ( x )$ to find a second approximation to $\beta$, giving your answer to 3 decimal places.
\hfill \mbox{\textit{Edexcel F1 2024 Q6 [8]}}